Bursting and Excitability in Neuromorphic Resonant Tunneling Diodes

We study in this paper the dynamics of quantum nanoelectronic resonant tunneling diodes (RTDs) as excitable neuromorphic spike generators. We disclose the mechanisms by which the RTD creates excitable all-or-nothing spikes and we identify a regime of bursting in which the RTD emits a random number of closely packed spikes. The control of the latter is paramount for applications in event-activated neuromorphic sensing and computing. Finally, we discuss a regime of multistability in which the RTD behaves as a memory. Our results can be extended to other devices exhibiting negative differential conductance.

Figure 1

Current-voltage nonlinear relationship of the double-barrier quantum-well resonant tunneling diode. The regions with positive differential conductance (PDC I, PDC II) and negative differential conductance (NDC) are also illustrated. Upper inset: symmetric double-barrier quantum-well nanostructure where $U_o$ is the potential barrier height, and $E_1$–$E_3$ are the resonant energy levels. Lower inset: schematic circuit of the RTD oscillator.

Figure 2

Stable and unstable solutions in equilibrium of Eqs. (2) and (3) under different parameters. The blue straight line and yellow dashed line correspond to the nullclines $\dot{V}=0$ and $\dot{I}=0$, respectively. Stable (unstable) fixed points are depicted as black (white) dots. Stable (unstable) periodic orbits are depicted as straight (dashed) red lines. (a) $R=1\mathrm{\Omega }$, $\mu =0.04{\mathrm{\Omega }}^{-1}$, $V_0=2.1\mathrm{V}$. (b) $R=10\mathrm{\Omega }$, $\mu =0.04{\mathrm{\Omega }}^{-1}$, $V_0=2.75\mathrm{V}$. (c) $R=10\mathrm{\Omega }$, $\mu =0.01{\mathrm{\Omega }}^{-1}$, $V_0=3.2\mathrm{V}$. (d) $R=60\mathrm{\Omega }$, $\mu =0.01{\mathrm{\Omega }}^{-1}$, $V_0=3.998\mathrm{V}.$

Figure 3

Periodic solution on the phase space for $R=0.01\mathrm{\Omega }$, $V_0=2.3\mathrm{V}$, and two different values for the stiffness coefficient: (a) $\mu =0.008{\mathrm{\Omega }}^{-1}$, (b) $\mu =0.001{\mathrm{\Omega }}^{-1}$. The $I$-$V$ curve (blue line) and load line (dashed orange line) are included. (c),(d) Evolution of over time of the output variables $V$ and $I$ corresponding to the periodic solutions in (a) and (b), respectively. The system exhibits stages of slow and fast dynamics when $\mu$ is small enough.

Figure 4

System’s response to several perturbations out of its natural equilibrium for parameters $R=0.1\mathrm{\Omega }$, $\mu =0.001{\mathrm{\Omega }}^{-1}$, $V_0=2.26\mathrm{V}$. (a) Trajectories on the phase plane $(V,I)$. (b) Evolution of variables $V$ and $I$ over time. Current perturbations under 0.0282 A induce an exponential decay (green curves and insets). Perturbations above this threshold lead to the generation of a single spike before returning to the attractor (red curves).

Figure 5

Numerical traces of output pulses randomly generated by perturbing with additive white noise at $R=0.01\mathrm{\Omega }$ and $\mu =0.001{\mathrm{\Omega }}^{-1}$. The system is biased in the first PDC region in (a) and (b) and in the second PDC region in (c). The input bias voltage and input noise intensity are (a) $V_0=2.26\mathrm{V}$, $\eta =0.009\mathrm{V}$, (b) $V_0=2.26\mathrm{V}$, $\eta =0.011\mathrm{V}$, (c) $V_0=2.94\mathrm{V}$, $\eta =0.014\mathrm{V}$.

Figure 6

(a) Bifurcation diagram illustrating the $I$ coordinate of the fixed point (in blue) and the $I$ maximum and minimum of the limit cycle (in red) versus the input bias voltage $V_0$ for $R=1\mathrm{\Omega }$ and $\mu =0.05{\mathrm{\Omega }}^{-1}$. Solid lines represent stable solutions and dashed lines represent unstable solutions. Andronov-Hopf bifurcations (AH) and limit-cycle folds ($F$) are also shown. (b) Evolution of the period $T$ of the limit cycle along the branch.

Figure 7

(a) Bifurcation diagram illustrating the $I$ coordinate of the fixed point (in blue) and the $I$ maximum and minimum of the limit cycle (in red) versus the input bias voltage $V_0$ for $R=1\mathrm{\Omega }$ and $\mu =0.002{\mathrm{\Omega }}^{-1}$. Solid lines represent stable solutions and dashed lines represent unstable solutions. The insets show an enlargement in the proximity to each AH bifurcation. The limit-cycle folds ($F$) are also shown. (b) Evolution of the period $T$ of the limit cycle along the branch.

Figure 8

Three-dimensional bifurcation diagram illustrating the stable steady-state branch (solid blue line), unstable steady-state branches (dashed blue lines), stable limit-cycle branch (colored gradient surface), unstable limit-cycle branches (red surfaces), and limit-cycle folds (red curves) versus the input bias voltage $V_0$ for $R=1\mathrm{\Omega }$ and different values of $\mu$: (a) $\mu =0.002{\mathrm{\Omega }}^{-1}$, (b) $\mu =0.05{\mathrm{\Omega }}^{-1}$, (c) $\mu =0.14{\mathrm{\Omega }}^{-1}$.

Figure 9

Andronov-Hopf bifurcation branches (blue line) and limit-cycle fold branches (red line) on the $\left(V_0,\mu \right)$ plane for $R=1\mathrm{\Omega }$ (a) and $R=10\mathrm{\Omega }$ (b). There is a unique stable fixed point (FP) outside the region delimited by the AH branch, which is unstable inside. In this region, there is a stable limit cycle (LC). In the regions between the AH branch and the limit-cycle fold branch, the system is bistable ($B$) in the sense that the stable fixed point and the stable limit cycle coexist.

Figure 10

Bifurcation diagram illustrating the $I$ coordinate of the fixed point (blue lines) as well as the $I$ maximum and minimum of the limit cycle (red lines) and homoclinic curves (brown dots) versus the input bias voltage $V_0$ for $R=60\mathrm{\Omega }$ and different values of $\mu$: (a) $\mu =0.006{\mathrm{\Omega }}^{-1}$, (b) $\mu =0.0115{\mathrm{\Omega }}^{-1}$, (c) $\mu =0.03{\mathrm{\Omega }}^{-1}$. Solid lines represent stable solutions and dashed lines represent unstable solutions. AH bifurcations, limit-cycle folds ($F$), saddle-node bifurcations (SN) and focus-node transitions (FN) are also illustrated. As $\mu$ increases, the AH bifurcations and focus-node transitions approach the saddle-node bifurcation, and a second pair of homoclinic curves arise. For $\mu >1/R$, there is no AH bifurcation and no periodic nor homoclinic solutions.

Figure 11

Three-dimensional bifurcation diagram illustrating the stable steady-state branch (solid blue line), unstable steady-state branches (dashed blue lines), stable limit-cycle branch (color gradient surface), unstable limit-cycle branches (red surfaces), limit-cycle folds (red curves), and homoclinic solutions (brown curves) versus the input bias voltage $V_0$ for $R=60\mathrm{\Omega }$ and different values of $\mu$: (a) $\mu =0.006{\mathrm{\Omega }}^{-1}$, (b) $\mu =0.0115{\mathrm{\Omega }}^{-1}$.

Figure 12

Homoclinic curves shown in Figs. 10 and 11 (brown straight line), arising for parameters $R=60\mathrm{\Omega }$, $\mu =0.0115{\mathrm{\Omega }}^{-1}$ and different values of $V_0$. (a) Type-1 homoclinic, $V_0=3.999\mathrm{V}$. (b) Type-2 homoclinic, $V_0=4.012\mathrm{V}$. (c) Type-3 homoclinic, $V_0=3.987\mathrm{V}$. (d) Type-4 homoclinic, $V_0=4.024\mathrm{V}$. The blue straight line and yellow dashed line correspond to the nullclines. Stable (unstable) fixed points are depicted as black (white) dots.

Figure 13

Bifurcation branches represented as curves on the $\left(V_0,\mu \right)$ plane for $R=60\mathrm{\Omega }$: Andronov-Hopf bifurcations (blue line), limit-cycle fold bifurcation (red line), saddle-stable node bifurcations (solid green line), saddle-unstable node bifurcations (dashed green line), type-1 homoclinic bifurcation (solid brown line), type-2 homoclinic bifurcation (dashed brown line), type-3 homoclinic bifurcation (dashed-dotted brown line), type-4 homoclinic bifurcation (dotted brown line), Bogdanov-Takens bifurcations (BT). The inset enlarges the black square in the figure, showing the homoclinic bifurcations in detail.

Figure 14

Stable solutions and the bifurcations that delimit them on the $\left(V_0,\mu \right)$ plane for $R=60\mathrm{\Omega }$. UA, upper attractor point; LA, lower attractor point; SC, stable limit cycle. 3S, coexistence of the three stable solutions.

Figure 15

Input bias-voltage range of the bistability region in terms of the resistance $R$ at the first and second PDC regions (red circles and blue squares, respectively). The values of the stiffness coefficient are $\mu =0.0002{\mathrm{\Omega }}^{-1}$ (empty symbols) and $\mu =0.002{\mathrm{\Omega }}^{-1}$ (solid symbols).

Figure 16

Period of the stable limit cycle versus the stiffness coefficient $\mu$, with the circuit biased in the first and second PDC (filled red and empty blue marks, respectively) for resistances $R=1\mathrm{\Omega }$ (circles) and $R=60\mathrm{\Omega }$ (squares). The dashed line has a slope -1 for reference.